The use of boundary locus plots in the identi cation ofbifurcation points in numerical approximation of delaydi

UMIST The use of boundary locus plots in the identiication of bifurcation points in numerical approximation of delay diierential equations Abstract We are interested in nonlinear delay diierential equations which have a Hopf bifurcation. We assume zero is a steady state for the problem, and so a Hopf bifurcation point lies on the boundary of the region of asymptotic stability for the zero solution. We investigate whether discrete versions of the nonlinear delay diierential equation also exhibit Hopf bifurcations. The analysis of Hopf bifurcation points can be mathematically complicated. Hopf bifurcation is an essentially nonlinear property and therefore the common practice of analysing stability of nonlinear problems through consideration of a linearised version does not apply. However some insights can nevertheless be obtained through a linear analysis and these are the subject of the present report. We use the boundary locus method as a tool to determine the boundary of the stability region, both for the zero solution of the delay diierential equation and for the zero solution of numerical analogues. We use the information obtained about the stability domain to assist in identifying Hopf bifurcations. We demonstrate the following: For certain linear multistep methods, the boundary of the region of stability for the solution of the original equation is approximated by the boundary of the region of stability for the zero solution of the numerical analogue equation to the order of the method. The boundary locus method enables us to determine precise parameter values at which any Hopf bifurcations arise in the discrete equations. We prove that Hopf bifurcation points for the true equation are approximated to the order of the method by corresponding points in the discrete scheme. Further calculations are necessary to determine the precise nature of bifurcation points identiied in this way.